For a tree $T$, the subtree polynomial of $T$ is the generating polynomial for the number of subtrees of $T$. We show that the complex roots of the subtree polynomial are contained in the disk $\left\{z\in ℂ:\left|z\right|\le 1+\sqrt[3]{3}\right\}$, and that $K_{1,3}$ is the only tree whose subtree polynomial has a root on the boundary. We also prove that the closure of the collection of all real roots of subtree polynomials contains the interval $[-2,-1]$, while the intervals $(-\infty ,-1-\sqrt[3]{3})$, $[-1,0)$, and $(0,\infty )$ are root-free.

中文翻译：

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在子树多项式的根上

对于一棵树 $Ť$，该子树多项式的$Ť$ 是的子树数的生成多项式 $Ť$。我们表明子树多项式的复数根包含在磁盘中$\left\{ž\in ℂ：\left|ž\right|\le \mathrm{1个}+\sqrt[3]{3}\right\}$， 然后 ${ķ}_{\mathrm{1个}，3}$是唯一其子树多项式在边界上具有根的树。我们还证明子树多项式的所有实根的集合的闭合包含区间$[-2，-\mathrm{1个}]$，而间隔 $（-\infty ，-\mathrm{1个}-\sqrt[3]{3}）$， $[-\mathrm{1个}，0）$和 $（0，\infty ）$ 是无根的。